Abstract
We introduce a new general polynomial-time construction- the fibre construction- which reduces any constraint satisfaction problem \({\rm CSP}(\mathcal H)\) to the constraint satisfaction problem \({\rm CSP}(\mathcal{P})\), where \({\mathcal{P}}\) is any subprojective relational structure. As a consequence we get a new proof (not using universal algebra) that \({\rm CSP}(\mathcal{P})\) is NP-complete for any subprojective (and thus also projective) relational structure. This provides a starting point for a new combinatorial approach to the NP-completeness part of the conjectured Dichotomy Classification of CSPs, which was previously obtained by algebraic methods. This approach is flexible enough to yield NP-completeness of coloring problems with large girth and bounded degree restrictions.
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Nešetřil, J., Siggers, M. (2007). Combinatorial Proof that Subprojective Constraint Satisfaction Problems are NP-Complete. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_16
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